Yep.
I'll only add that to take it one step furhter, if I may want more room ambience and/or crowd reaction that I'd get from onstage alone, I'll add a pair onstage facing out into the room and audience (poiniting away from the stuff on stage and the PA, sometimes even under the stage-lip to help block that stuff if there aren't subs under there and not blocked by bodies directly in front). Takes either more than 4 channels, or recording a single channel mono SBD and a single channel mono room/aud mic to do it with 4 channels though. I do it with the 6 channels of a DR680 and either run 3 pairs (2 OnStage, 2 SBD, 2 Room), or (3 on stage as L/C/R, 1 SBD, 2 room) if the SBD is mono. Sometimes mono SBD and one close mic onstage for acoustic bass or something.
That provides control over balancing all three improtant elements in the mix: the on-stage sound for image and instumental detail, the direct SBD sound for clarity and vocals, and the room/audience facing mics for ambience/audience reaction. An AUD from farther back in the room is a mediocre compromise of all those things, even the room ambience since it is time delayed, and especially audience reaction since the less interested folks chatting with each other tends to hang in back, while the visceral thrill of those totally into it is up in front! If more audience reaction helps the mix, that's the kind I want.